Archive for November 25th, 2011

Group actions III — what’s the point of them?

November 25, 2011

Somebody told me recently that a few years ago they had a supervision with a colleague of mine (who shall remain nameless, but he or she is an applied mathematician) and asked what the point of group actions was. “I have absolutely no idea,” was the response, and the implication that one might draw from it was apparently intended.

No pure mathematician could hold such a view. I’ve stated a few times that group actions tell you a lot about groups. In this post I want to try to explain why that is, though there is far more to say than I am capable of explaining, let alone fitting into one blog post.

Several proofs that use group actions seem to depend on almost magically coming up with an action that just happens, when you analyse it the right way, to tell you what you wanted to know. I am not an algebraist and do not have a good all-purpose method for finding actions to prove given statements. I don’t rule out that such a method might exist, at least for reasonably simple statements, and would be interested to hear from anybody who thinks they can usefully add to what I have to say.